Page - 409 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 425 ThecurvaturetensorR∈T 31 (M)givesthegl(n)-valuedcurvatureformΩ :TFM×TFM→gl(n) onTFMby Ω(vu,wu)=u−1R(π∗(vu),π∗(wu))u , vu,wv∈TFM . Note thatΩ(vu,wu) =Ω(hu(π∗(vu)),hu(π∗(wu))), whichwe canuse towrite the curvatureR as the gl(n)-valuedmap Ru : T2(Tπ(u)M)→ gl(n), (v,w) →Ω(hu(π∗(vu)),hu(π∗(wu))) for fixed u. Incoordinates, thecurvature is R sijk =Γ l ikΓ s jl−ΓljkΓsil+Γsik;j−Γsjk;i whereΓsik;j= ∂xjΓ s ik. Letxt,sbea familyofpaths inM, and letut,s∈π−1(xt,s)behorizontal liftsofxt,s foreachfixed s. Write x˙t,s = ∂txt,s and u˙t,s = ∂tut,s. The s-derivative ofut,s canbe regardedapushforwardof the horizontal lift andis thecurve inTFM ∂sut,s=ψ ( ut,s,ψ−1u0,s(C(∂su0,s))+ ∫ s 0 Ω(u˙r,s,∂sur,s)dr ) +hut,s(∂sxt,s) =ψ ( ut,s,ψ−10,s(C(∂su0,s))+ ∫ s 0 Rur,s(x˙r,s,∂sxr,s)dr ) +hut,s(∂sxt,s) . (2) This follows from the structure equation dω =−ω∧ω+Ω (e.g., [21]). Note that the curve dependson theverticalvariationC(∂su0,s)atonlyonepointalong thecurve. Theremaining terms dependon thehorizontal variationor, equivalently, ∂sxt,s. The t-derivativeof ∂sut,s is the curve in TTFM satisfying ∂shut,s(x˙t,s)=ψ ( ut,s,Rut,s(x˙t,s,∂sxt,s) ) +∂tψ ( ut,s,ψ−10,s(C(∂su0,s)) ) +∂t ( hut,s(∂sxt,s) ) =ψ ( ut,s,Rut,s(x˙t,s,∂sxt,s) ) +∂tψ ( ut,s,ψ−10,s(C(∂su0,s)) ) +hut,s(∂t∂sxt,s)+(∂thut,s)(∂sxt,s). (3) Here, thefirstandthird terminthe lastexpressionare identifiedwithelementsofT∂sut,sTFMby thenaturalmappingTut,sFM→T∂sut,sTFM.WhenC(∂su0,s) iszero, therelationreflects theproperty that thecurvatureariseswhencomputingbracketsbetweenhorizontalvectorfields.Note that thefirst termof (3)hasvalues inVFM,while the third termhasvalues inHFM. 3.TheAnisotropicallyWeightedMetric For a Euclidean driftless diffusion process with spatially constant stochastic generator Σ, the log-probabilityofasamplepathcanformallybewritten ln p˜Σ(xt)∝− ∫ 1 0 ‖x˙t‖2Σdt+cΣ (4) with thenorm‖·‖Σgivenbythe innerproduct 〈v,w〉Σ= 〈 Σ−1/2v,Σ−1/2w 〉 = vΣ−1w; i.e., the inner productweightedbytheprecisionmatrixΣ−1. Thoughonlyformal,as thesamplepathsarealmost surelynowheredifferentiable, the interpretationcanbegivenaprecisemeaningbytaking limitsof piecewise linearcurves [21]. Turning to themanifoldsituationwith theprocessesmappedtoMby stochasticdevelopment, theprobabilityofobservingadifferentiablepathcaneitherbegivenaprecise meaning in themanifoldby taking limitsof small tubesaroundthecurve, or inRd byconsidering infinitesimal tubesaroundtheanti-developmentof thecurves.With the former formulation,ascalar curvaturecorrection termmustbeaddedto (4),givingtheOnsager–Machlupfunction[22]. The latter formulationcorresponds todefininganotionofpathdensity for thedrivingRd-valuedprocessWt. WhenM isRiemannianandΣunitary, takingthemaximumof(4)givesgeodesicsasmostprobable paths for thedrivingprocess. 409